0132_1302_2031_2130
Counting sequence:
1, 1, 2, 6, 20, 66, 213, 676, 2121, 6601, 20419, 62861, 192772, 589271, 1796480, 5464476, 16589858, 50283816, 152197731, 460117786, 1389580839, 4192900351, 12641947057, 38091421331, 114707758450, 345258178541, 1038745750718, 3124015994226, 9392412993776, 28230555272526, 84831249680049, 254859184336336, 765534275035077, 2299107555887221, 6903880137981535, 20728808094122201, 62231369852579488, 186811778588199491, 560743397285768636, 1683036707390355336, 5051221607249044094, 15159192761448017076, 45492050618368727487, 136514040917479326406, 409641317605532734755, 1229183648313509326891, 3688230836578166592973, 11066472489150504493151, 33204077514061889009806, 99624432702600788906201, 298905238542437356818554, 896799336770801918878686, 2690616933308240311805132, 8072423947768734752499546, 24218772363842411153581965, 72660245505291840331911676, 217991021236633134713148993, 653999989458407638420523041, 1962070460899990004387383291, 5886395934525763046258516261, 17659670966529901148938833724, 52980277836621746444209989551, 158944145158008756314647148792, 476841105481424776892599498356, 1430546014818326337541755470810, 4291697469569736525268479287616, 12875247985679430106871119512603, 38626151262834194401167190611826, 115879520128920074914097386407711, 347641352102216795793525770865751, 1042931365112602608823682332952809, 3128813230040207919749128448282411, 9386489785421871817638524672763562, 28159600507466959534810910552615621, 84479144880703662791727601932905270, 253438333565817996855172080089566026, 760317354110272231818189192866190248, 2280958223645564410732597162100196966, 6842890801468118136779207284207977321, 20528714634683881408803839662844464936, 61586254464358800317228756621287607229, 184759052843718342225671764951904925901, 554277916322773694408154542487086877847, 1662835732892535145143895875267334829681, 4988512392658628953458845121588854977240, 14965550775994743352538575604321042200091, 44896687928059776016074690035839707917108, 134690156986387109431438919536594390433376, 404070714965709126485502343674132390029366, 1212212783714562992646848936786369559593676, 3636640023589448019320386942586676628568439, 10909924449288185568910339318678761345561886, 32729784810978322198198713296564910466467867, 98189384443756421558050047420361879228893571, 294568231900619864073044169512558454745368325, 883704901399089935452360682761426667582677751, 2651115242719610236657872543704061095729063878, 7953347138028621657642760943147775262783710641, 23860045105172897385634922270130320622159658226, 71580144978909998447355541820416354392308034486, 214740460235816881800712311050638245920602756484
Generating function in Maple syntax:
(x^5-5*x^4+16*x^3-17*x^2+7*x-1)/(3*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{5}-5 x^{4}+16 x^{3}-17 x^{2}+7 x -1}{\left(3 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 5*x**4 + 16*x**3 - 17*x**2 + 7*x - 1)/((x - 1)**2*(3*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(3*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)-x^5+5*x^4-16*x^3+17*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{5}+5 x^{4}-16 x^{3}+17 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/60*(6*5^(1/2)-30)*(3/2-1/2*5^(1/2))^(-n)+1/60*(-6*5^(1/2)-30)*(3/2+1/2*5^(1/2))^(-n)-1/2*n+5/12*3^n+5/4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 \sqrt{5}-30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(-6 \sqrt{5}-30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}-\frac{n}{2}+\frac{5 \,3^{n}}{12}+\frac{5}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(n+3) = 3*a(n)-10*a(n+1)+6*a(n+2)-n+2, n >= 6
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 66
a \! \left(n +3\right) = 3 a \! \left(n \right)-10 a \! \left(n +1\right)+6 a \! \left(n +2\right)-n +2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15655/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[12,x]*F[4,x]
F[4,x] = F[31,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[20,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[30,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[24,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[12,x]*F[13,x]
F[31,x] = F[32,x]+F[46,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[1,x]+F[14,x]
F[36,x] = F[32,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[32,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[40,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[43,x]
F[46,x] = F[21,x]+F[47,x]+F[75,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[57,x]
F[49,x] = F[50,x]+F[6,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[24,x]+F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[51,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[53,x]
F[57,x] = F[46,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[61,x]+F[72,x]
F[61,x] = F[62,x]+F[70,x]
F[62,x] = F[21,x]+F[63,x]+F[68,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[10,x]+F[54,x]
F[66,x] = F[62,x]+F[67,x]
F[67,x] = F[59,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[32,x]+F[62,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[61,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[74,x]
F[74,x] = F[12,x]*F[72,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[69,x]+F[77,x]
F[77,x] = F[37,x]+F[78,x]
F[78,x] = 2*F[21,x]+F[79,x]+F[84,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[61,x]+F[81,x]
F[81,x] = F[78,x]+F[82,x]
F[82,x] = F[83,x]
F[83,x] = F[12,x]*F[81,x]
F[84,x] = F[12,x]*F[77,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{31}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{20}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{46}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)+F_{47}\! \left(x \right)+F_{75}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{57}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{6}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{57}\! \left(x \right) = F_{46}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{72}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{70}\! \left(x \right)
F_{62}\! \left(x \right) = F_{21}\! \left(x \right)+F_{63}\! \left(x \right)+F_{68}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{10}\! \left(x \right)+F_{54}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{59}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \left(x \right)+F_{62}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{69}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{37}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{79}\! \left(x \right)+F_{84}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{61}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{78}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_31(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_24(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_12(x)*F_13(x))
Eq(F_31(x), F_32(x) + F_46(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_1(x) + F_14(x))
Eq(F_36(x), F_32(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_32(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_40(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_43(x))
Eq(F_46(x), F_21(x) + F_47(x) + F_75(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_57(x))
Eq(F_49(x), F_50(x) + F_6(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_24(x) + F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_51(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_53(x))
Eq(F_57(x), F_46(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_72(x))
Eq(F_61(x), F_62(x) + F_70(x))
Eq(F_62(x), F_21(x) + F_63(x) + F_68(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_10(x) + F_54(x))
Eq(F_66(x), F_62(x) + F_67(x))
Eq(F_67(x), F_59(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_32(x) + F_62(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_61(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_12(x)*F_72(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_69(x) + F_77(x))
Eq(F_77(x), F_37(x) + F_78(x))
Eq(F_78(x), 2*F_21(x) + F_79(x) + F_84(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_61(x) + F_81(x))
Eq(F_81(x), F_78(x) + F_82(x))
Eq(F_82(x), F_83(x))
Eq(F_83(x), F_12(x)*F_81(x))
Eq(F_84(x), F_12(x)*F_77(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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